By plane (from Greece to Italy)
​For a faster and more comfortable trip you can use a flight to Thessaloniki or Athens airport
Daily flights are available with Aegean airlines or low cost companies Ryannair airlines, or Volotea

From Napoli to Matera and back
Autobus from Napoli airport you can go to Napoli Centrale /Metropark Staz.FS Terminalbus (bus station), after you can take Flixbus or MarinoBus
(autobus "Alibus" from Airport to Napoli Centrale every 15 minutes more or less in front of "Arrivi" doors) 
 
From Rome to Napoli 
From Rome airport ( Fumicino or Ciampino) you can take the airport bus  to Rome centrale and then a train to Napoli using trenitalia
From Napoli Centrale /Metropark Staz.FS Terminalbus (bus station), after you can take Flixbus or MarinoBus
(autobus "Alibus" from Airport to Napoli Centrale every 15 minutes more or less in front of "Arrivi" doors) 

From Bari airport to Matera centrale and return
You can use the bus 
or
From Bari Centrale/Fal station to Matera Centrale
https://ferrovieappulolucane.it/ (bus or train, it depends on the day or on the time) 

From Bari airport to Bari Centrale: bus linea 16 AMTAB or train FR2 (grigia) and FM" (gialla) 

By Car
You can also rent a car from Napoli airport using the platform carjet